7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²...

Question

7–84. Evaluate the following integrals.

59. ∫ 1/(x⁴ + x²) dx

Accepted Answer

Welcome back everyone. In this problem, we want to evaluate the integral of 1 divided by Xquared multiplied by X2 + 4 with respect to X. Now to help us evaluate this interval, it would be a good idea for us to set up partial fractions, OK? And that will make our term easier to integrate. So by doing that What we're seeing is that we want to rewrite 1 divided by X2 multiplied by X2 + 4, OK, in terms of some constant divided by X, some constant B divided by X2, plus some constant C multiplied by X plus D divided by X2 + 4. How can we do that? Well, first, let's multiply both sides by X, multiplied by X2 + 4. When we do that, we end up with 1 being equal to AX multiplied by X2 + 4, plus B multiplied by X2 + 4, plus C X + D multiplied by X2. If we expand, then we're gonna have AX cubed plus 4 A X, OK, plus B X squared plus 4B. Plus C X cubed plus DX2. I know when we group our like terms according to our variables, then we're gonna have a plus C multiplied by X cub plus B + D multiplied by X2 + 4 A X + 4 B all being equal to 1. Now if we were to compare, comparing both sides of the equation. And comparing both sides of the equation. Notice here that only on the left hand side of the equation there's only a constant, and on the right hand side of the equation our constant is 4 B. So what does this tell us? Well, here this tells us then that if we compare X cubed terms, A plus C would be equal to 0 since there is no X cubed term. B + D would also be equal to 0 because there's no X2 term. For A would be equal to 0 because there's no X term, and 4B would be equal to 1. Now, if 4B equals 1, that means B must be equal to a quarter. And if B equals a quarter, remember that B plus D equals 0, so that means D must be equal to a negative quarter. If 4 A equals 0, then A equals 0. And if A equals 0, then C must be equal to 0, because A plus C is 0. So what we're saying then. Is that 1 divided by x2 multiplied by X2 + 4 split into partial fractions would be a 0 divided by X plus B quarter divided by X2 plus C X + D, that's 0 X plus negative quarter divided by X2 + 4. And if we simplify that, it's going to be equal to 1 divided by 4 X2 minus 1 divided by 4 multiplied by X2 + 4. So this means that we can rewrite our integral, OK. As 1 divided by 4 X squad minus 1 divided by 4 multiplied by X2 + 4, all with respect to X. Now, where do we go from here? Well, we can integrate each term because by what using what we know about integration, we could rewrite this as a quarter of the integral of 1 divided by Xquad with respect to X and our other term. As a minus quarter of the integral of 1 divided by X2 + 4 with respect to X. So let's go ahead and integrate each term, OK? So now we're integrating each term. Now, starting with our first term, OK. A quarter of the integral of 1 divided by X squared with respect to X is going to be equal to -1 divided by 4 X plus some constant term C1. Now for our next term, that is a quarter of the integral of 1 divided by x squared plus 4 with respect to x. Notice that we could rewrite this as a quarter of the integral of 1 divided by X2 + 2 squad with respect to x. Now why would we do that? Well, if we recall for the standard form and integration, recall that one, sorry, the integral of 1 divided by, let me get rid of this box, OK, because it kind of makes it a little harder to write, but we're saying the integral of 1 divided by X2 plus a squared with respect to X is going to be equal to 1 divided by A. Multiplied by the arc tangent of X divided by a plus some constant C. So if we use that idea, then we're saying that the integral of our term here where A is going to be equal to 2 is going to be equal to a quarter multiplied by 1/2 of the arc tangent of X divided by 2 plus some constant C2. And when we simplify that, that's gonna be 1/8 of the arc tangent of a half of X plus C2. Now that we've integrated both terms, we can combine the results. And now by doing that, OK, then the integral of what we originally started with. OK. Is going to be equal to -1 divided by 4 X minus 1/8 of the arc tangent of X divided by 2 plus C, where C is a combination of our two constants C1 and C2. So thus this is our final answer. This is our integral. Thanks a lot for watching everyone. I hope this video helped.

7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²) dx

Key Concepts

Integration

Partial Fraction Decomposition

Polynomial Factorization

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